Bayesian inverse problems (BIPs) governed by large-scale complex models in high parameter dimensions (such as PDEs with discretized infinite-dimensional parameter fields) quickly become prohibitive, since the forward model must be solved numerous times---as many as millions---to characterize the uncertainty in the parameters.
Efficient evaluation of the parameter-to-observable (p2o) map, which involves solution of the forward model, is key to making BIPs
tractable. Surrogate approximations of p2o maps have the potential to greatly accelerate BIPs, provided that the p2o map can be accurately approximated using (far) fewer forward model solves than would be required for solving the BIP using the full p2o map. Unfortunately, constructing such surrogates presents significant challenges when the parameter dimension is high and the forward model is expensive.
Deep neural networks (DNNs) have emerged as leading contenders for overcoming these challenges. We demonstrate that black box application of DNNs for problems with infinite dimensional parameter fields leads to poor results, particularly in the common situation when training data are limited due to the expense of the model. However, by constructing a network architecture that is adapted to the geometry and intrinsic low-dimensionality of the p2o map as revealed through adjoint PDEs, one can construct a dimension-independent "reduced basis" DNN surrogate with superior generalization properties using only limited training data. We employ this reduced basis DNN surrogate to make tractable the solution of Bayesian optimal experimental design problems, in particular for finding sensor locations that maximize the expected information gain. Application to inverse wave scattering is presented.
This work is joint with Tom O'Leary-Roseberry, Keyi Wu, and Peng Chen.